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Full text of "Treatise On Analysis Vol-Ii"

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3 HAUSDORFF SPACES 7 n exists a neighborhood Vf of b which does not contain at, and V = f) Vf is then a neighborhood of Z> which does not contain any of the a{. (12.3.5) Let f, g be two continuous mappings of a topological space E into a Hausdorff space E. Then the set A of points xeE such thatf(x) = g(x) is closed inE. The proof is similar to that of (3.15.1) (which is a special case of (12.3.5)). If a $ A, then f(d) =£ g(a), and so there are disjoint neighborhoods U' of f(d) and V ofg(d). Since/"'(U') and/'^V) are neighborhoods of a in E, the same is true of their intersection W, and it is clear that/(^:) ^ g(x) for all x e W. Hence E — A is open. The principle of extension of identities (3.15.2) therefore remains true for continuous mappings of any topological space into a Hausdorff space. Proposition (3.15.3) and its corollary (3.15.4) (the "principle of extension of inequalities ") are also valid—the proofs are the same—for arbitrary topologi- cal spaces. (12.3.6) Let E be a compact metrizable space, F a Hausdorff space, and fa continuous mapping ofE into F. Thenf(E) is closed in F. Iff is infective, it is a homeomorphism ofE onto the subspacef(E) ofE. Let y be a point of the complement of/(E). For each z e/(E), there exist disjoint open neighborhoods V(z) of z and W(z) of y. The inverse images /~1(V(z)) form an open covering of E as z runs through/(E) (3.11.4). Hence by compactness there exists a finite number of points zt e/(E) such that the V(z£) form an open covering of/(E) in F. But then U = P) W(zf) is an open i neighborhood of y which does not intersect /(E). This shows that /(E) is closed in F. It follows that if A is any closed subset ofE, then/(A) is closed in F (and therefore also in/(E)), because A is compact (3.17.3). If/is injective, this establishes the continuity of the inverse mapping/(E) -»E (3.11.4). (12.3.7) If a Hausdorff topology is coarser than the topology of a compact metrizable space, then the two topologies coincide. (12.3.8) Let E be a compact metrizable space, F a Hausdorff space, f a con- tinuous mapping ofE into F. If b is any point off(E) and U is any open set in E containing /"*(#), then there exists a neighborhood V of b in F such thated is said to be obtained by patching